Codensity and the ultrafilter monad

Tom Leinster

Even a functor without an adjoint induces a monad, namely, its
codensity monad; this is subject only to the existence of certain
limits. We clarify the sense in which codensity monads act as
substitutes for monads induced by adjunctions. We also expand on
an undeservedly ignored theorem of Kennison and Gildenhuys: that
the codensity monad of the inclusion of (finite sets) into (sets)
is the ultrafilter monad. This result is analogous to the
correspondence between measures and integrals. So, for example,
we can speak of integration against an ultrafilter. Using this
language, we show that the codensity monad of the inclusion of
(finite-dimensional vector spaces) into (vector spaces) is double
dualization. From this it follows that compact Hausdorff spaces
have a linear analogue: linearly compact vector spaces.
Finally, we show that ultraproducts are categorically
inevitable: the codensity monad of the inclusion of (finite
families of sets) into (families of sets) is the ultraproduct
monad.